Find the equation of the common tangent to the circles x ^ 2 + y ^ 2 = 25 and (x - 8) ^ 2 + y ^ 2 = 25

To find the equation of the common tangent to the circles:

1. **Equations of the Circles:**
   - Circle 1: \( x^2 + y^2 = 25 \) (center \( C_1(0, 0) \), radius \( r_1 = 5 \))
   - Circle 2: \( (x - 8)^2 + y^2 = 25 \) (center \( C_2(8, 0) \), radius \( r_2 = 5 \))

The centers of the two circles are \( C_1(0, 0) \) and \( C_2(8, 0) \), and both have radii of 5.

### Step 1: Distance between the centers
The distance between the centers \( C_1(0, 0) \) and \( C_2(8, 0) \) is:
\[
d = \sqrt{(8 - 0)^2 + (0 - 0)^2} = 8
\]
Since the radii of both circles are 5 and \( d = 8 \), the circles are separated and have two common external tangents.

### Step 2: Equation of the common tangent
For two circles with equal radii \( r_1 = r_2 = r \) and centers separated by a distance \( d \), the equation of the common external tangent can be derived using the formula:

\[
y = mx + c
\]
For tangents to circles, the formula for the common tangent is:
\[
c = \pm \frac{\sqrt{d^2 - (r_1 + r_2)^2}}{2}
\]
However, since the radii are equal and the circles are symmetric, the equation simplifies to using geometry to find the slope, as they will intersect

Find the equation of the common tangent to the circles x^2 + y^2 = 25 and (x - 8)^2 + y^2 = 25.

Learn how to find the equation of the common tangent to the circles x^2 + y^2 = 25 and (x - 8)^2 + y^2 = 25. This problem covers key geometric concepts such as the distance between points, tangent lines, and properties of circles. Ideal for high school students in Grades 10-12.

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